Electromagnetics and Calculation of Fields

Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Nathan Ida Joao P.A. Bastos

Electromagnetics and Calculation of Fields Second Edition

With 500 Illustrations

i Springer

Nathan Ida Department of Electrical Engineering The University of Akron Akron, OH 44325-3904, USA

Consulting Editor R. Mittra, Ph.D.

Joao P.A. Bastos Department of Electrical Engineering Federal University of Santa Catarina Florianopolis, SC 880549, Brazil

Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801-2991, USA

Library of Congress Cataloging-in-Publication Data Ida, Nathan.

Electromagnetics and calculation of fields / Nathan Ida, loao P.A. Bastos. - 2nd ed.

p. cm. Includes bibliographical references and index. ISBN 0-387-94877-5 (alk. paper) I. Electromagnetism. 2. Electromagnetic fields. 1. Bastos,

Joao. QC760.156 1997 537 - dc20 96-42356

Printed on acid-free paper.

© 1997 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf­ter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Lesley Poliner; manufacturing supervised by Jeffrey Taub. Camera-ready copy prepared from the authors' MS Word files.

987654321

ISBN 0-387-94877-5 Springer-Verlag New York Berlin Heidelberg SPIN 10552732

Preface

This second edition of Electromagnetics and Calculation of Fields was undertaken with the intention of both updating the first edition and, most importantly, introducing the new subject of edge or vector finite elements. We felt that edge elements have matured enough to be included in a text of this type. In addition, the original text was expanded to include an introduction to the finite element method as a general design tool with in-depth discussion of the method's practical aspects, before its application to specific electromagnetic phenomena is introduced. Many of the features of the previous edition as well as the general appearance and flavor were retained while the coverage was extended and the presentation improved.

The text is intended as an introduction to electromagnetics and computation of electromagnetic fields. While many texts on electromagnetics exist, the subject of computation of electromagnetic fields is normally not treated or is treated in a number of idealized examples, with the main emphasis on development of theoretical relations.

"Why another book on electromagnetics?" This is perhaps the first question the reader may ask when opening this book. It is a valid question, because among the many books on electromagnetics some are excellent. We have two answers to this question, answers that have motivated the writing of this book.

The first concerns the method of presentation of electromagnetism. Generally, in classical books the material is presented in the following sequence: electrostatics, magnetostatics, magnetodynamics, and wave propagation, using integral forms of the field equations. As a primary effect of this presentation, the reader is led to think that the knowledge of this science is synonymous with memorizing dozens of formulas. Additionally, an impression that there is no firm connection between these equations lingers in the reader's mind since at each step new postulates are added, seemingly unrelated to previous material. Our opinion is, and we shall try to convey this to the reader, that the electromagnetic formalism is extremely simple and based on very few equations. They are the four "Maxwell equations" which include practically all the existent relationships between the electromagnetic quantities. The only additional relationships that need be considered is the Lorentz force and the material constitutive relations.

Maxwell's four equations are presented in the point, or local form. This treatment provides a powerful tool, because it is valid for any situation and is independent of geometry. When we wish to apply them to a real device, it is more practical to use the integral form of the equations, because applications lead quite

v

VI Preface

naturally to the notions of line, surface, and volume. In this formalism, the "dozens of formulae" associated with the classical methods of presentation are particular cases of Maxwell's equations as applied to specific geometries. Thus, it seems to us to be more useful to present the Electromagnetic theory based on a few equations, together with the tools needed for their general use. We will show that the electrostatic, magnetostatic, and magnetodynamic applications are particular cases of these four equations. Under certain conditions, Maxwell's equations can be decoupled into two principal groups (electrostatic and magnetic) which can be studied, and used, independently. However, while the electromagnetic formalism is simple, it does not mean that all its phenomena are necessarily easy to solve. There are many complex problems whose solutions are the subject of extensive current inquiry. However, this does not prevent one from trying to present the Electromagnetic theory in a simple, generally applicable way.

Many maintain that this is a science with a definite theoretical character. This is certainly true, but we will try to emphasize its applied aspects associated with a variety of applications. However, it is not an exercise book. There are many good books dedicated exclusively to problem solving. The final sections of most chapters contain a few, carefully chosen examples. These are presented with comments, in order to complement and simplify the understanding of the theoretical material presented in the chapter.

The facts discussed above have motivated the writing of the first part of the book, titled "The Electromagnetic Field and Maxwell's Equations." This part is divided into seven chapters.

In the first chapter we present the mathematical basis necessary to work with the Maxwell's equations in integral and differential forms.

Chapter 2 describes the common basis of electromagnetics, or Maxwell's equations, with special attention to the conditions under which they can be decoupled into two independent groups of equations: one for electrostatic and the other for magnetic phenomena.

In Chapter 3, the electrostatic field equations are studied. The electric field intensity, dielectric strength, and capacitance are presented. In its final section, the finite element method is described, with the goal of proposing a solution of the Laplace equation related to the electric field.

The fourth chapter treats the second group of the decoupled Maxwell equations: the magnetostatic field equations. Here the static aspects of magnetism is presented. The Biot-Savart law, magnetic materials, the notion of inductance, etc. are presented. Permanent magnets are discussed in detail, because of their importance in design of high-performance magnetic devices.

Chapter 5 deals with magnetodynamic phenomena. Field penetration in conductors, eddy currents, eddy current losses, hysteresis losses, and induced currents due to movement are discussed.

In Chapter 6, we discuss the interactions between electromagnetic and mechanical quantities. Among these, we mention here the force on conductors, force on charges (Lorentz force), magnetic field energy, and the Poynting vector.

Preface Vll

Special attention is given to Maxwell's force tensor. This aspect of electromagnetic fields is rarely presented in classical books. It is very useful, because it permits, in a simple and elegant way, the calculation of the force exerted on a body by the magnetic field on the surface enclosing the body. We have a particular interest in calculation of forces using Maxwell's force tensor because of its usefulness in numerical calculation of forces.

Chapter 7 deals with what we call high-frequency electromagnetic fields and wave propagation aspects. The distinction between low and high-frequencies is not on frequency but rather on displacement currents. High-frequency applications are those in which displacement currents cannot be neglected, regardless of frequency. Wave propagation, scattering, reflection, transmission, and refraction of plane waves are discussed first, followed by fields in waveguides and cavity resonators. Propagation in dielectrics, lossy dielectrics, and conductors are also presented.

The second answer to the question posed above relates to the second part of the book, entitled "Introduction to the Finite Element Method in Electromagnetics."

The analysis and design of an electrical device requires the knowledge of the electromagnetic phenomena throughout the device. The difficulty of solving Laplace's and Poisson's equations for electrical and magnetic fields was an enormous obstacle in achieving this important goal for a long time. The solution of second order partial differential equations is a special domain in applied mathematics. Methods of solution were limited to a few simple cases until very recently. By 1950, the finite elements method began to emerge as a general method for solution of mechanical engineering problems. Around 1970 this method began to be adapted to electrical engineering applications. The methods employed prior to this, either empirical or approximate, were not satisfactory, especially when the structure had a complex geometry. The use of the finite element method opened a new era in applied electromagnetism and, as a consequence, in electrical engineering, which had important needs in computer­aided design tools. The knowledge offield distribution permits the rational design construction of structures efficiently, economically, and within safety standards.

One of the principal goals of this work is to present the Finite Element method and its application to electromagnetic problems. The didactic approach used here is quite different than in other books on the subject: we present the problems to be solved and then the application of the finite elements to these problems. This means that the principal goal of this book is to solve electromagnetic problems, and the use of finite elements technique is the tool of choice in achieving this end. In this sense, this part of the book can be viewed as a tutorial on problem solving using the finite element method. Although problem solving is the major aspect, the material is presented in detailed fashion to allow the reader to fOlm a firm basis in this important aspects of applied electromagnetics.

The second part of the book is divided into six chapters. The main objective of this part is the didactic presentation of the finite element method which by now has

Vlll Preface

become an indispensable method for analysis and design of electromagnetic devices.

In Chapter 8, we introduce the finite element method as a general engineering design tool in preparation for specific applications in the following chapters. The concepts associated with the finite element method are discussed in depth and are accompanied by simple examples. A small finite element program for second order elements is included to show how the concepts are linked to practical implementation.

In Chapter 9, we present the variational method. This theoretical formulation, associated with finite elements, constitutes a well-adapted method to solve static problems. The axi-symmetric formulation, the Newton-Raphson method (for nonlinear problems) as well as several examples are described in this chapter.

Chapter 10 introduces Galerkin's residual method as it relates to time dependent applications. Two-dimensional (2D) and three-dimensional (3D) finite elements and their application are shown.

Chapter 11 introduces the concept of edge elements. The hexahedral edge element shape functions are presented in a didactic manner and subroutines showing how they are constructed are listed. The application of edge elements to static 3D and eddy current problems is explained and graphical results are provided to show their utility.

In Chapter 12, we describe several computational aspects associated with the implementation of Finite Element programs, with special attention to the matrix system, to memory use, and to the solution methods needed to solve the system of equations obtained. Iterative, direct, and eigenvalue problems are discussed.

In the thirteenth and last chapter, many aspects of the general organization of field calculation software are discussed. Pre- and post-processing methods (such as mesh generation and equipotential lines drawing) are described. The more modern systems (extension of traditional software packages) and trends in this research field are also discussed in the final sections of this chapter.

Finally, we would like to thank R. Carlson, B. Davat, R. Mesquita, G. Meunier, A. Raizer, C. Rioux, and N. Sadowski who have, in different ways, contributed to the realization of this work. Special thanks goes to many others, and in particular students, who pointed out errors, misspellings, and possible ways of improving the presentation.

N. Ida J. P. A. Bastos November 1996

Contents

Preface

Part I. The Electromagnetic Field and Maxwell's Equations

1. Mathematical Preliminaries 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

1.2. The Vector Notation. . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1.3. Vector Derivation .... . . . . . . . . . . . . . . . . . . . . . . .. 2

1.3.1. The Nabla ('\1) Operator. . . . . . . . . . . . . . . . . . . .. 2 1.3.2. Definition of the Gradient, Divergence, and Curl . . . . .. 2

1.4. The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 1.4.1. Example of Gradient ...................... 5

1.5. The Divergence ............................. 6 1.5.1. Definition of Flux .. . . . . . . . . . . . . . . . . . . . . .. 6 1.5.2. The Divergence Theorem . . . . . . . . . . . . . . . . . . .. 8 1.5.3. Conservative Flux . . . . . . . . . . . . . . . . . . . . . . .. 9 1.5.4. Example of Divergence . . . . . . . . . . . . . . . . . . . .. 11

1.6. The Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12 1.6.1. Circulation of a Vector ...... . . . . . . . . . . . . . .. 12 1.6.2. Stokes'Theorem . . . . . . . . . . . . . . . . . . . . . . . .. 14 1.6.3. Example of Curl . . . . . . . . . . . . . . . . . . . . . . . .. 17

1.7. Second Order Operators ........................ 18 1.8. Application of Operators to More than One Function ....... 19 1.9. Expressions in Cylindrical and Spherical Coordinates . . . . . .. 20

2. The Electromagnetic Field and Maxwell's Equations 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22 2.2. Maxwell's Equations .......................... 22

2.2.1. Fundamental Physical Principles of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23

ix

x Contents

2.2.2. Point Form of the Equations ......... . 2.2.3. The Equations in Vacuum .......... . 2.2.4. The Equations in Media with £=E{) and j..l=J.l{)

2.2.5. The Equations in General Media 2.2.6. The Integral Form of Maxwell's Equations

2.3. Approximations to Maxwell's Equations 2.4. Units .......................... .

3. Electrostatic Fields 3.1. Introduction . . . . . . . . . . . .... . 3.2. The Electrostatic Charge ... . .... .

3.2.1. The Electric Field ................. . 3.2.2. Force on an Electric Charge ..... . 3.2.3. The Electric Scalar Potential V ... .

29 32 34 35 37 43 45

47 47 48 48 49

3.3. Nonconservative Fields: Electromotive Force 53 3.4. Refraction of the Electric Field 55 3.5. Dielectric Strength. . . . . . . . . 59 3.6. The Capacitor ........... . . . . . . . 61

3.6.1. Definition of Capacitance . . . . . . . 61 3.6.2. Energy Stored in a Capacitor. . . . . . . . . . . . . . . . .. 64 3.6.3. Energy in a Static, Conservative Field ............ 64

3.7. Laplace's and Poisson's Equations in Terms of the Electric Field 65 3.8. Examples .................... 67

3.8.1. The Infinite Charged Line ...... 67 3.8.2. The Charged Spherical Half-Shell 70 3.8.3. The Spherical Capacitor ....... 71 3.8.4. The Spherical Capacitor with Two Dielectric Layers 72

3.9. A Brief Introduction to the Finite Element Method: Solution of the Two-Dimensional Laplace Equation ........ . . . . . . . .. 74 3.9.1. The Finite Element Technique for Division of a Domain . 75 3.9.2. The Variational Method .............. 77 3.9.3. A Finite Element Program .. . . . . . . . . . . . . . . . .. 80 3.9.4. Example for Use of the Finite Element Program ...... 84

3.10. Tables of Permittivities, Dielectric Strength, and Conductivities 88

4. Magnetostatic Fields 4.1. Introduction . . . . . . . . . . . . . . . . . . . . 90 4.2. Maxwell's Equations in Magnetostatics 91

4.2.1. The Equation V'xH=J 91

4.2.2. The Equation V'·B=O 93

4.2.3. The Equation V'xE=O 93

Contents Xl

4.3. The Biot-Savart Law .............. . 94 4.4. Boundary Conditions for the Magnetic Field . 96 4.5. Magnetic Materials ....... 98

4.5.1. Diamagnetic Materials . 99 4.5.2. Paramagnetic Materials . 100 4.5.3. Ferromagnetic Materials 100 4.5.4. Permanent Magnets .. . . . . . . . . 104

4.6. The Analogy between Magnetic and Electric Circuits . . . . 115 4.7. Inductance and Mutual Inductance . . . . . . . . . 119

4.7.1. Definition ofInductance ......... . 4.7.2. Energy in a Linear System ............ . 4.7.3. The Energy Stored in the Magnetic Field ... .

4.8. Examples .......................... . 4.8.1. Calculation of Field Intensity and Inductance of a Long

Solenoid ................ . 4.8.2. Calculation of H for a Circular Loop ........ . 4.8.3. Field of a Rectangular Loop '" .......... . 4.8.4. Calculation of Inductance of a Coaxial Cable ... . 4.8.5. Calculation of the Field Inside a Cylindrical Conductor .. 4.8.6. Calculation of the Magnetic Field Intensity in a Magnetic

Circuit .............................. . 4.8.7. Calculation of the Magnetic Field Intensity of a Saturated

Magnetic Circuit . . . . . . . . . . . . . . . . . . . . . . . 4.8.8. Magnetic Circuit Incorporating Permanent Magnets ..

4.9. Laplace's Equation in Terms of the Magnetic Scalar Potential 4.10. Properties of Soft Magnetic Materials ............. .

5. Magnetodynamic Fields 5.1. Introduction ............................... . 5.2. Maxwell's Equations for the Magnetodynamic Field ....... . 5.3. Penetration of Time Dependent Fields in Conducting Materials .

5.3.1. The Equation for H 5.3.2. The Equation for B . 5.3.3. The Equation for E ........... . 5.3.4. The Equation for J .. 5.3.5. Solution of the Equations

5.4. Eddy Current Losses in Plates . 5.5. 5.6.

Hysteresis Losses ....... . Examples ............ . 5.6.1. Induced Currents Due to Change in Induction .. 5.6.2. Induced Currents Due to Changes in Geometry . 5.6.3. Inductive Heating of a Conducting Block ....

119 .120 122 123

123 125 127 128 129

130

133 135 138 140

142 143 146 146 147 147 148 148 153 156 160 160 163 165

xii Contents

5.6.4. Effect of Movement of a Magnet Relative to a Flat Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 169

5.6.5. Visualization of Penetration of Fields as a Function of Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 171

5.6.6. Thc Voltage Transformer. . . . . . . . . . . . . . . . . . .. 172

6. Interaction between Electromagnetic and Mechanical Forces 6.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175 6.2. Force on a Conductor . . . . . . . . . . . . . . . . . . . . . . . . .. 175 6.3. Force on Moving Charges: The Lorentz Force ........... 178 6.4. Energy in the Magnetic Field ..................... 180 6.5. Force as Variation of Energy (Virtual Work) ............ 182 6.6. The Poynting Vector .......................... 184 6.7. Maxwell's Force Tensor ........................ 188 6.8. Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.8.1. Force between Two Conducting Segments ......... 195 6.8.2. Torque on a Loop ........................ 198 6.8.3. The Hall Effect .......................... 200 6.8.4. The Linear Motor and Generator ............... 202 6.8.5. Attraction of a Ferromagnetic Body ............. 205 6.8.6. Repulsion of a Diamagnetic Body . . . . . . . . . . . . . . . 206 6.8.7. Magnetic Levitation. . . . . . . . . . . . . . . . . . . 207 6.8.8. The Magnetic Brake ....................... 209

7. Wave Propagation and High-Frequency Electromagnetic Fields 7.1. Introduction ................................ 212 7.2. The Wave Equation and Its Solution ................. 215

7.2.1. The Time Dependent Equations . . . . . . . . . . . . . . . . 215 7.2.2. The Time Harmonic Wave Equations ............ 220 7.2.3. Solution of the Wave Equation ................ 222 7.2.4. Solution for Plane Waves .................... 222 7.2.5. The One-Dimensional Wave Equation in Free Space and

Lossless Dielectrics . . . . . . . . . . . . . . . . . . . . . . . 223 7.3. Propagation of Waves in Materials .................. 227

7.3 1. Propagation of Waves in Lossy Dielectrics ......... 227 7.3.2. Propagation of Plane Waves in Low-Loss Dielectrics ... 229 7.3.3. Propagation of Plane Waves in Conductors ......... 230 7.3.4. Propagation in a Conductor: Definition of the Skin Depth 232

7.4. Polarization of Plane Waves ...................... 233

Contents xiii

7.5. Reflection, Refraction, and Transmission of Plane Waves .... 235 7.5.1. Reflection and Transmission at a Lossy Dielectric Interface:

Normal Incidence ........................ 236 7.5.2. Reflection and Transmission at a Conductor Interface:

Normal Incidence ........................ 239 7.5.3. Reflection and Transmission at a Finite Conductivity

Conductor Interface . . . . . . . . . . . . . . . . . . . . . . . 240 7.5.4. Reflection and Transmission at an Interface:

Oblique Incidence . . . . . . . . . . . . . . . . . . . . . . . . 241 7.5.5. Oblique Incidence on a Conducting Interface:

Perpendicular Polarization . . . . . . . . . . . . . . . . . . . 242 7.5.6. Oblique Incidence on a Conducting Interface:

Parallel Polarization . . . . . . . . . . . . . . . . . . . . . . . 244 7.5.7. Oblique Incidence on a Dielectric Interface:

Perpendicular Polarization . . . . . . . . . . . . . . . . . . . 245 7.5.8. Oblique Incidence on a Dielectric Interface:

Parallel Polarization . . . . . . . . . . . . . . . . . . . . . . . 248 7.6. Waveguides ................................ 249

7.6.1. TEM, TE, and TM Waves ................... 249 7.6.2. TEM Waves ........................... 251 7.6.3. TE Waves ............................. 251 7.6.4. TM Waves ............................ 252 7.6.5. RectangularWaveguides .................... 253 7.6.6. TM Modes in Waveguides ................... 253 7.6.7. TE Modes in Waveguides ................... 256

7.7. Cavity Resonators ............................ 258 7.7.1. TM and TE Modes in Cavity Resonators . . . . . . . . . . . 259 7.7.2. TE Modes in a Cavity . . . . . . . . . . . . . . . . . . . . . . 261 7.7.3. Energy in a Cavity ........................ 261 7.7.4. Quality Factor of a Cavity Resonator . . . . . . . . . . . . . 263 7.7.5. Coupling to Cavities ....................... 263

Part II. Introduction to the Finite Element Method in Electromagnetics

8. Introduction to the Finite Element Method 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 8.2. The Galerkin Method - Basic Concepts ....... . . . . . . . . 266 8.3. The Galerkin Method - Extension to 2D . . . . . . . . . . . . . . . 277

8.3.1. The Boundary Conditions ................... 278 8.3.2. Calculation of the 2D Elemental Matrix . . . . . . . . . . . 279

8.4. The Variational Method - Basic Concepts .............. 281

XIV Contents

8.5. The Variational Method - Extension to 2D 8.5.1. The Variational Fonnulation ....... . 8.5.2. Calculation of the 2D Elemental Matrix .... .

8.6. Generalization of the Finite Element Method . 8.6.1. High-Order Finite Elements: General ...... . 8.6.2. High-Order Finite Elements: Notation ..... . 8.6.3. High-Order Finite Elements: Implementation .. 8.6.4. Continuity of Finite Elements .......... . 8.6.5. Polynomial Basis ............... . 8.6.6. Transfonnation of Quantities - the Jacobian 8.6.7. Evaluation of the Integrals

8.7. Numerical Integration ............... . 8.7.1. Evaluation of the Integrals ........ . 8.7.2. Basic Principles of Numerical Integration 8.7.3. Accuracy and Errors in Numerical Integration

8.8. Some Specific Finite Elements . 8.8.1. 1D Elements 8.8.2. 2D Elements ....... .

284 284 289 291 292 293 296 298 298 300 302 306 306 307 311 313 314 315

8.8.3. 3D Elements ........ 318 8.9. Coupling Different Finite Elements; Infinite Elements 323

8.9.1. Coupling Different Types of Finite Elements 323 8.9.2. Infinite Elements. . . . . . . . . . . . . . . . . . . 325

8.10. Calculation of Some Terms in Poisson's Equation .. 327 8.10.1. The Stiffness Matrix. . . . . . . . . . . . . . . . 327 8.10.2. Evaluation of the Second Tenn in Eq. (8.130) 329 8.10.3. Evaluation of the Third Tenn in Eq. (8.130) . . 329 8.10.4. Evaluation of the Source Term .......... 330

8.11. A Simplified 2D Second-Order Finite Element Program 330 8.11.1. The Problem to Be Solved . . . 330 8.11.2. The Discretized Domain .. 332 8.11.3. The Finite Element Program . . 333

9. The Variational Finite Element Method: Some Static Applications 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . 343 9.2. Some Static Applications. . . . . . . . . . . . . 343

9.2.1. Electrostatic Fields: Dielectric Materials 343 9.2.2. Stationary Currents: Conducting Materials 345 9.2.3. Magnetic Fields: Scalar Potential . . . . 346 9.2.4. The Magnetic Field: Vector Potential. . 347 9.2.5. The Electric Vector Potential .. 352

9.3. The Variational Method ............. 353

Contents xv

9.3.1. The Variational Formulation. . . . . . . . . . 354 9.3.2. Functionals Involving Scalar Potentials .. 355 9.3.3. The Vector Potential Functionals . . . . . . . . . . . 359

9.4. The Finite Element Method ................... 362 9.5. Application of Finite Elements with the Variational Method 366

9.5.1. Application to the Electrostatic Field ........ 367 9.5.2. Application to the Case of Stationary Currents ... 370 9.5.3. Application to the Magnetic Field: Scalar Potential 370 9.5.4. Application to the Magnetic Field: Vector Potential 371 9.5.5. Application to the Electric Vector Potential . . 373

9.6. Assembly of the Matrix System . 373 9.7. Axi-Symmetric Applications. . . . . . . . . . 375 9.8. Nonlinear Applications . . . . . . . . . . . . . 383

9.8.1. Method of Successive Approximation 383 9.8.2. The Newton-Raphson Method . . . . 384

9.9. The Three-Dimensional Scalar Potential . . . 387 9.9.1. The First-Order Tetrahedral Element . 388 9.9.2. Application of the Variational Method 389 9.9.3. Modeling of 3D Permanent Magnets 389

9.10. Examples . . . . . . . . . . . . . . . . . . . 390 9.10.1. Calculation of Electrostatic Fields 391 9.10.2. Calculation of Static Currents ......... 392 9.10.3. Calculation of the Magnetic Field: Scalar Potential 394 9.10.4. Calculation of the Magnetic Field: Vector Potential 396 9.10.5. Three-Dimensional Calculation of Fields of Permanent

Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

10. Galerkin's Residual Method: Applications to Dynamic Fields

10.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 400 10.2. Application to Magnetic Fields in Anisotropic Media 401 10.3. Application to 2D Eddy Current Problems . . . . . . 405

10.3.1. First-Order Element in Local Coordinates . . 405 10.3.2. The Vector Potential Equation Using Time

Discretization . . . . . . . . . . . . . . . . . 409 10.3.3. The Complex Vector Potential Equation ... 417 10.3.4. Structures with Moving Parts ................ 420 10.3.5. The Axi-Symmetric Formulation .............. 422 10.3.6. A Modified Complex Vector Potential Formulation for

Wave Propagation . . . . . . . . . . . . . . . . . . . 425 10.3.7. Formulation of Helmholtz's Equation ............ 427 10.3.8. Advantages and Limitations of 2D Formulations . . . 430

10.4. Application of the Newton-Raphson Method ........... 432

XVI Contents

10.5. Examples ................................ 434 lO.5.1. Eddy Currents: Time Discretization ............ 434 lO.5.2. Moving Conducting Piece in Front of an

Electromagnet ......................... 437 lO.5.3. Modes and Fields in a Waveguide ............. 440 10.5.4. Resonant Frequencies of a Microwave Cavity ...... 442

11. Hexahedral Edge Elements - Some 3D Applications

11.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 11.2. The Hexahedral Edge Element Shape Functions ......... 448 11.3. Construction of the Shape Functions . . . . . . . . . . . . . . . . 456 11.4. Application of Edge Elements to Low-Frequency

Maxwell's Equations ......................... 460 11.4.1. Static Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 11.4.2. Listing of the Matrix Construction Code . . . . . . . . . . 464 11.4.3. Modeling of Permanent Magnets .............. 466 11.4.4. Eddy Currents - the Time-Stepping Procedure . . . . . . 466

11.4.5. Eddy Currents - The Complex Formulation . . . . . . . . 468 11.4.6. The Newton-Raphson Method . . . . . . . . . . . . . . . . 469 11.4.7. The Divergence of J and Other Particulars ........ 471

11.5. Modeling of Waveguides and Cavity Resonators ......... 472 11.6. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

11.6.1. Static Calculations (TEAM Problem 13) .......... 474 11.6.2. A Linear Motor with Permanent Magnets .. . . . . . . . 475 11.6.3. Eddy Current Calculations (TEAM Problem 21) ..... 477 11.6.4. Calculation of Resonant Frequencies

(TEAM Problem 19) ..................... 480

12. Computational Aspects in Finite Element Software Implementation

12.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 484 12.2. Geometric Repetition of Domains .................. 484

12.2.1. Periodicity ........................... 484 12.2.2. Anti-Periodicity ........................ 486

12.3. Storage of the Coefficient Matrix . . . . . . . . . . . . . . . . . . 487 12.3.1. Symmetry of the Coefficient Matrix ............ 487 12.3.2. The Banded Matrix and Its Storage ............. 487 12.3.3. Compact Storage of the Matrix ............... 489

12.4. Insertion of Dirichlet Boundary Conditions . . . . . . . . . . . . 490 12.5. Quadrilateral and Hexahedral Elements .............. 491

Contents xvii

12.6. Methods of Solution of the Linear System . . . . . . . . . . . . . 493 12.6.1. Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . 493 12.6.2. Iterative Methods ....................... 497

12.7. Methods of Solution for Eigenvalues and Eigenvectors ..... 500 12.7.1. The Jacobi Transformation .................. 500 12.7.2. The Givens Transformation ................. 503 12.7.3. The QR and QZ Methods ................... 504

12.8. Diagram of a Finite Element Program ............... 506

13. General Organization ofField Computation Software

13.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 13.2. The Pre-Processor Module ...................... 510

13.2.1. The User/System Dialogue .................. 510 13.2.2. Domain Discretization .................... 511

13.3. The Processor Module ........................ 516 13.4. The Post-Processor Module ..................... 517

13.4.1. Visualization of Results ................... 517 13.4.2. Calculation of Numerical Results . . . . . . . . . . . . . . 519

13.5. The Computational Organization of a Software Package .... 524 13.5.1. The EFCAD Software ..................... 525

13.6. Evolving Software . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 13.6.1. The Adaptive Mesh Method ................. 528 13.6.2. A Coupled Therma1JElectrical System ........... 533 13.6.3. A Software Package for Electrical Machines ....... 537 13.6.4 A System for Simultaneous Solution of Field Equations

and External Circuits ..................... 541 13.6.5. Computational Difficulties and Extensions to Field

Computation Packages .................... 547 13.7. Recent Trends ............................. 547

Bibliography ................................. 549

Subject Index ................................. 559